Definition:Orthogonal Polynomials

Definition

Let $S$ be a set of polynomials over the real numbers $\R$:

$S = \set {\map {p_0} x, \map {p_1} x, \map {p_2} x, \ldots}$

such that $p_n$ is of degree $n$.

The elements of $S$ are orthogonal with respect to a closed real interval $\closedint a b$ and a continuous non-negative weight function $\map w x$ on $\closedint a b$ if and only if:

$\ds \int_a^b \map w x \map {p_i} x \map {p_j} x \rd x = 0$

for all $i \ne j$.

Examples

Chebyshev Polynomials

The Chebyshev polynomials of the first kind form a set of orthogonal polynomials with respect to:

the closed real interval $\closedint {-1} 1$

the weight function $\map w x := \dfrac 1 {\sqrt {1 - x^2} }$ on $\closedint {-1} 1$

Also see

• Results about orthogonal polynomials can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): orthogonal polynomials
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): orthogonal polynomials